Abstract
Planar motion of a non-deformable wheel under the action of non-ideal unilateral constraints is considered. The mathematical description of this phenomenon has a form of a non-smooth initial value problem. The non-smoothness of this problem means that its solution is determined by an absolutely continuous function having a discontinuous first derivative. For this reason, a collision problem describing abrupt changes of velocity has been formulated next to the equations of motion specifying the acceleration. The non-idealness of constraints means that the constraint reaction force includes also a component resulting from the friction between the wheel and the constraints. Differential equations specifying acceleration of the wheel making contact with the constraints and algebraic equations for determining the changes in the wheel’s velocity at the moment of collision have been formulated in the paper. The principal task in these formulations is to determine the reaction forces of the considered constraints. This task is specified by the relationships between acceleration and the constraint reaction force components. In the description of the collision, these relations refer to the post-collision velocities and reaction force impulses. For determining an approximate solution of the formulated wheel motion problem, an original numerical method and a computer program for wheel motion simulation have been developed. Selected results illustrating the changes in displacements and velocity have been presented.
Highlights
The non-ideal constraints model so specified is described by relations, which determine the relationship between the reaction force R and the acceleration and velocity of the point A belonging to the wheel: Figure 4
The non-smooth problem of a mechanical system motion was closely analyzed in the monograph [1], where perfect unilateral constraints were considered
An analysis of the non-smooth problem of the motion of a mechanical system in the form of a wheel with constrained motion has been presented in the paper
Summary
N degrees of freedom, we find—within a fixed time interval—the solution to the differential equation,. In the monograph [1] where a system with unilateral constraints is analysed it is demonstrated that the solution to the problem exists within the class of absolutely continuous functions ( ) [ ) X ∈ Cab to ,tend , RN. This means that the derivative X is discontinuous, and, an additional task specifying discontinuous changes in velocity needs to be formulated. In the further considered non-smooth wheel motion problem, we investigate non-ideal unilateral constraints. In addition to formulating the motion problem, we propose a method for determining the solution to this problem and present the results of wheel motion simulations
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